Visualizing and Experimenting with Knot Invariants Through 3D Modeling Techniques

Abstract

Background: Knot theory studies the embeddings of the closed curves inside a three-dimensional space and various invariants of knots such as the Alexander polynomial, Jones polynomial, Seifert genus, and hyperbolic volume are used to distinguish different types of knots. Such invariants play a pivotal role in many scientific disciplines, such as quantum physics, molecular biology and computer science. The problem is that the algebra rules are realityually quite abstract and can be difficult for students to visualize.

Objective: In this work we utilize 3D modeling to render and empirically study knot invariants in order to increase the insight of - and engagement with - aspects that relate to knot theory.

Method: Classical knots such as trefoil (3₁), figure-eight (4₁), and cinquefoil (5₁) were modeled in Blender, and analyzed by using SnapPy and custom Python scripts. The numerical values of the Jones polynomial V(q), Alexander polynomial Δ(t), the Seifert genus g, the Euler characteristic χ, and hyperbolic volume V_hyp were computed. Knot isotopies are simulated to examine invariance under Reidemeister moves. The topological properties were also cross-verified by visual inspection as well as the algebraic calculation.

Results The computationally derived invariant value of each model was very close to the known values, which demonstrates the 3D models of torus knot to be reliable. For instance, the Jones polynomial for 3₁ was proved to be V(q) = q + q³ - q⁴ and the hyperbolic volume for 4₁ was established to be approximately V_hyp ≈ 2.02988. The topological stability under deformation was supported by visual experimentation.

Conclusion: By enabling 3D models to be used in combination with numerical topology, we obtain a powerful tool for the dynamic, intuitive study of knot invariants, providing an interface between theoretical computation and experimental visualization.